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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 45968.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45968.t1 | 45968k4 | \([0, -1, 0, -305608, 45037680]\) | \(159661140625/48275138\) | \(954428909873733632\) | \([2]\) | \(663552\) | \(2.1551\) | |
45968.t2 | 45968k3 | \([0, -1, 0, -278568, 56675696]\) | \(120920208625/19652\) | \(388532021116928\) | \([2]\) | \(331776\) | \(1.8085\) | |
45968.t3 | 45968k2 | \([0, -1, 0, -116328, -15229072]\) | \(8805624625/2312\) | \(45709649543168\) | \([2]\) | \(221184\) | \(1.6058\) | |
45968.t4 | 45968k1 | \([0, -1, 0, -8168, -173200]\) | \(3048625/1088\) | \(21510423314432\) | \([2]\) | \(110592\) | \(1.2592\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 45968.t have rank \(0\).
Complex multiplication
The elliptic curves in class 45968.t do not have complex multiplication.Modular form 45968.2.a.t
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.